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Properties

Label 2-630-35.27-c1-0-14
Degree $2$
Conductor $630$
Sign $0.535 - 0.844i$
Analytic cond. $5.03057$
Root an. cond. $2.24289$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (1.52 + 1.63i)5-s + (2.33 − 1.23i)7-s + (−0.707 + 0.707i)8-s + (−0.0743 + 2.23i)10-s + 5.73·11-s + (−3.41 − 3.41i)13-s + (2.52 + 0.781i)14-s − 1.00·16-s + (2.57 − 2.57i)17-s − 1.85·19-s + (−1.63 + 1.52i)20-s + (4.05 + 4.05i)22-s + (−6.46 + 6.46i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.683 + 0.730i)5-s + (0.884 − 0.466i)7-s + (−0.250 + 0.250i)8-s + (−0.0234 + 0.706i)10-s + 1.72·11-s + (−0.946 − 0.946i)13-s + (0.675 + 0.208i)14-s − 0.250·16-s + (0.624 − 0.624i)17-s − 0.424·19-s + (−0.365 + 0.341i)20-s + (0.864 + 0.864i)22-s + (−1.34 + 1.34i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.535 - 0.844i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.535 - 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10152 + 1.15615i\)
\(L(\frac12)\) \(\approx\) \(2.10152 + 1.15615i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (-1.52 - 1.63i)T \)
7 \( 1 + (-2.33 + 1.23i)T \)
good11 \( 1 - 5.73T + 11T^{2} \)
13 \( 1 + (3.41 + 3.41i)T + 13iT^{2} \)
17 \( 1 + (-2.57 + 2.57i)T - 17iT^{2} \)
19 \( 1 + 1.85T + 19T^{2} \)
23 \( 1 + (6.46 - 6.46i)T - 23iT^{2} \)
29 \( 1 - 3.47iT - 29T^{2} \)
31 \( 1 + 0.469iT - 31T^{2} \)
37 \( 1 + (-0.574 - 0.574i)T + 37iT^{2} \)
41 \( 1 + 1.03iT - 41T^{2} \)
43 \( 1 + (-6.17 + 6.17i)T - 43iT^{2} \)
47 \( 1 + (3.85 - 3.85i)T - 47iT^{2} \)
53 \( 1 + (-1.85 + 1.85i)T - 53iT^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 - 8.53iT - 61T^{2} \)
67 \( 1 + (4.46 + 4.46i)T + 67iT^{2} \)
71 \( 1 + 5.13T + 71T^{2} \)
73 \( 1 + (7.80 + 7.80i)T + 73iT^{2} \)
79 \( 1 + 4.16iT - 79T^{2} \)
83 \( 1 + (-1.77 - 1.77i)T + 83iT^{2} \)
89 \( 1 - 9.12T + 89T^{2} \)
97 \( 1 + (-0.0119 + 0.0119i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.74095406273984254032980261843, −9.864349638423290574226746037466, −9.034442699404014707860791109564, −7.71798115416903589611621685938, −7.23943341991003289896924106989, −6.16985740808422592994539135622, −5.38131311479610563436005749064, −4.24305094545786786877288557409, −3.18127295273226963847190692954, −1.69343461731589521466703354349, 1.45302494088623170524881812365, 2.28499887245081888306289350857, 4.15364496848223703105821280580, 4.63371448625823558299080429943, 5.88104951366655530589371861149, 6.51243694157367193368616025360, 8.011778886964802840014850194265, 8.949134830701367124070514388558, 9.558317455994286798858381592933, 10.45787782081760361654643497789

Graph of the $Z$-function along the critical line